# How to Safely Convert Between Units

We can convert from **km/h** (kilometers per hour) to **m/s** (meters per second) like this:

A kilometer has 1000 meters, and an hour has 3600 seconds, so a kilometer per hour is:

- \frac{1000}{3600} = 0.2777... m/s

How did I know to make it \frac{1000}{3600} and not \frac{3600}{1000} (the other way around)?

The trick is to do the conversions as fractions!

### Example 1:

Let's start with a simple example: **convert 3 km to m** (3 kilometers to meters). There are 1000 m in 1 km, so the conversion is easy, but let's follow a system.

The system is:

- Write the conversion as a fraction that equals 1
- Multiply it out (leaving all units in the answer)
- Cancel any units that are both top and bottom

We can write the conversion as a **fraction that equals 1**:

\frac{1000 m}{1 km} = 1

And it is safe to multiply by 1 (does not affect the answer):

3 km × 1 = 3 km

so we can do this:

3 km × \frac{1000 m}{1 km} = \frac{3000 km · m}{ 1 km}

The answer looks strange! But we aren't finished yet ... we can "cancel" any units that are both top and bottom:

\frac{3000 km · m}{1 km} = 3000 m

So, 3 km equals 3000 m. Well we knew that, but we want to follow a **system**, so that when things get harder we know what to do!

And when we do it correctly we get to cancel units that are both top and bottom, and get a neat answer.

Note: if we do it **wrong** (with the conversion upside down) we get this:

3 km × \frac{1 km}{1000 m} = \frac{3 km · km}{1000 m}

And that doesn't let us do any cancelling!

### Example 2:

Let's use this method to solve the **km/h** to **m/s** conversion from the top of the page.

We will do it in two stages:

- from km/h (kilometers per hour) to m/h (meters per hour), then
- from m/h (meters per hour) to m/s (meters per second).

#### 1. From km/h (kilometers per hour) to m/h (meters per hour)

\frac{1 km}{ h} × \frac{1000 m}{1 km} = \frac{1000 km · m}{1 h · km}

Now "cancel out" any units that are both top and bottom:

\frac{1000 km · m}{1 h · km} = \frac{1000 m}{ 1 h}

It is now in meters per hour.

#### 2. From m/h (meters per hour) to m/s (meters per second)

To go from m/h (meters per hour) to m/s (meters per second) we put the "3600 seconds in an hour" conversion "upside down" because we want an "h" on top (so they will cancel later) :

\frac{1000 m}{ 1 h} × \frac{1 h}{3600 s} = \frac{1000 m · h}{3600 h · s}

Then "cancel out" any units that are both top and bottom:

\frac{1000 m · h}{3600 h · s } = \frac{1000 m}{ 3600 s}

And so our anwer is:

\frac{1000 m}{3600 s} = **0.2777... m/s**

Doing it **wrong** (with the the 3600 seconds/hour the other way around) gets this:

\frac{1000 m}{1 h} × \frac{3600 s}{1 h} = \frac{1000 × 3600 m · s}{1 h · h}

And there is nothing to cancel!

So we know we made a mistake, and can correct it.

#### All In One Go

With experience we can do it in one line like this:

\frac{1 km}{1 h} x \frac{1000 m}{1 km} × \frac{1 h}{3600 s}

= \frac{1000 km · m · h}{3600 h · km · s}

= \frac{1000 m}{3600 s}

Or even "all in one go" (crossing out as we go) like this:

\frac{1 km}{1 h} x \frac{1000 m}{1 km} × \frac{1 h}{3600 s} = \frac{1000 m}{3600 s}

### Example 3

Now let's use this method to do a real-world conversion.

What is 60 mph (miles per hour) in m/s (meters per second) ?

\frac{60 mile}{h} × \frac{1609 m}{mile} × \frac{1 h}{3600 s}

= \frac{60 × 1609 mile · m · h}{3600 h · mile · s}

= 26.82 m/s

## Summary

The important points are:

- Write the conversion as a fraction (that equals one)
- Multiply it out (leaving all units in the answer)
- Cancel any units that are both top and bottom